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Special cases of linear mappings (a) Rotations around the origin Let f be the counterclockwise rotation by the angle ϕ around the zero point (0; 0) (origin of the cartesian coordinate system). Each vector is rotated by ϕ, its image has the same length as before. Image vectors of the standard basis vectors = ? cos ϕ cos ϕ sin ϕ – sin ϕ We obtain as image vectors: The matrix of f is thus: We call this a rotation matrix. 97 cosϕ sin ϕ and − sin ϕ . cos ϕ (b) Scaling Let f be the mapping which enlarges (or shrinks) every vector by a certain, fixed factor λ ≠ 0. Its corresponding matrix is , called a scaling matrix with factor λ. Indeed, we have The image of each vector has the same direction as before (or the opposite direction, if λ is negative), but a length which is modified by the factor |λ|. Parallelism and all angles remain unchanged under this mapping. 98 (c) Centroaffine mapping Let f act as a scaling by λ1 on the x axis and as a scaling by λ2 on the y axis, with two different numbers λ1 ≠ λ2. The corresponding matrix is This mapping is called centroaffine. Its effect: It works as a pure scaling on the x and y axis, but not for vectors which are outside these coordinate axes (they are also rotated a bit): λ λ The centroaffine mapping is thus not a scaling for all vectors. Certain vectors play a special role for this mapping, namely, those on the coordinate axes: They are only scaled, the others are also rotated. 99 11. Eigenvalues and eigenvectors We have seen in the last chapter: for the centroaffine mapping , some directions, namely, the directions of the coordinate axes: 10 and 10 , are distinguished among all directions in the plane: In them, f acts as a pure scaling. We want to generalize this to arbitrary linear mappings. We call a vector representing such a direction an eigenvector of the linear mapping f (or of the corresponding matrix A), and the scaling factor which describes the effect of f on it an eigenvalue. Examples: is eigenvector of the matrix to the eigenvalue 3: is also eigenvector of to the eigenvalue 3: 100 is eigenvector of to the eigenvalue 7: in general: An eigenvector of A must fulfill require . , and we Definition: Let A be a matrix of type (n, n). If there exists a real number λ such that the equation has a solution , we call λ an eigenvalue and an eigenvector of the matrix A. r A⋅ x r x If is an eigenvector of A and a ≠ 0 an arbitrary factor, then also is an eigenvector of A. We can choose a in a way that the length of becomes 1. That means, we can always find eigenvectors of length 1. 101 If we insert , we can transform the equation in the following way: This is equivalent to a system of linear equations with matrix A – λE and with right-hand side always zero. If the matrix A – λE has maximal rank (i.e., if it is regular), this system has exactly one solution (i.e., the trivial solution: the zero vector). We are not interested in that solution! The system has other solutions (infinitely many ones), if and only if A – λE is singular, that means, if and only if det(A – λE) = 0. From this, we can derive a method to determine all eigenvalues and eigenvectors of a given matrix. The equation det(A – λE) = 0 (called the characteristic equation of A) is an equation between numbers (not vectors) and includes the unknown λ. Solving it for λ means finding all possible eigenvalues of A. 102 In the case of a 2×2 matrix A, the characteristic equation det(A – λE) = 0 has the form i.e., it is a quadratic equation and can be solved with the well-known pq formula (see Chapter 6, p. 28). Example: is called the characteristic polynomial of A. Its zeros, the solutions eigenvalues of A. , are the 103 That means: Exactly for and does the vector equation have nontrivial solution vectors , i.e., eigenvectors. The next step is to find these eigenvectors vor each of the eigenvalues: This means to solve a system of linear equations! We use the equivalent form . We are not interested in the trivial solution In the example: To find an eigenvector to the eigenvalue (system of 2 linear equations with r.h.s. 0) 104 r r x = 0. with elementary row operations we get: From the second-last row we deduce: We can choose one parameter arbitrarily, e.g., x2 = c , and obtain the general solution (with c ∈ IR and c ≠ 0 because we want to have an eigenvector) It is enough to give just one vector as a representative of this direction, e.g., This is an eigenvector of A to the eigenvalue 1/2. Test: The eigenvectors to the second eigenvalue, 3/2, are determined analogously (a solution is −11 .) 105 In the general case of an n×n matrix, det(A – λE) is a polynomial in the variable λ of degree n, i.e., when we develop the determinant, we get something of the form Such a polynomial has at most n zeros, so A can have at most n different eigenvalues. Attention: There are matrices which have no (real) eigenvalues at all! Example: Rotation matrices with angle ϕ ≠ 0°, 180°. It is also possible that for the same eigenvalue, there are different eigenvectors with different directions. Example: For the scaling matrix vector , every is eigenvector to the eigenvalue 5. Fixed points and attractors Let f: IRn → IRn be an arbitrary mapping. r x ∈ IRn is called a fixed point of f, if r x i.e., if remains "fixed" under the mapping f. 106 , r x is called attracting fixed point, point attractor or vortex point of f , if there exists additionally a r r y neighbourhood of x such that for each from this neighbourhood the sequence r x converges against . The fixed points of linear mappings are exactly (by definition) the eigenvectors to the eigenvalue 1 and the zero vector. Examples: (shear mapping): each point on the x axis is a fixed point. (scaling by 2): only the origin (0; 0) is fixed point. (There are no eigenvectors to the eigenvalue 1; the only eigenvalue is 2.) The origin is not attracting. (scaling by 1/2, i.e., shrinking): the origin (0; 0) is attracting fixed point. 107 Definition: A stochastic matrix is an n×n matrix where all columns sum up to 1. Theorem: Each stochastic matrix has the eigenvalue 1. The corresponding linear mapping has thus a fixed point . Example from epidemiology: The outbreak of a disease is conceived as a stochastic (random) process. For a tree there are two possible states: "healthy" (state 0) and "infected" (state 1). For a healthy tree, let us assume a probability of 1/4 to be infected after one year, i.e.: , and correspondingly: (= probability to stay healthy). For infected trees, we assume a probability of spontaneous recovery of 1/3: We define the transition matrix (similar to the ageclasses example) as 108 For the purpose of calculation, we need the transposed of P, which is a stochastic matrix (and is in the literature also often called the transition matrix): A process of this sort, where the probability to come into a new state depends only on the current state, is called a Markov chain. Graphical representation of the transitions: "infected" If we assume that g1, resp., k1 are the proportions of healthy, resp., infected trees in the first year, the average proportions in the 2nd year are given by: 109 Question: what is the percentage of infected trees, if the tree stand is undisturbed for many years and the transition probabilities remain the same? We have to look for a fixed point of the mapping corresponding to PT. Because PT is a stochastic matrix, it has automatically the eigenvalue 1. We have only to determine a corresponding eigenvector (fixed point) gk '' : By applying the standard method for solving linear systems, we obtain: . From this we derive the proportion of the infected trees: Remarks: This proportion does not depend on the number of infected trees in the first year. g ' k' is in fact an attracting fixed point, if we restrict ourselves to a fixed total number of trees, g+k. 110 In the same way, a stable age-class distribution can be calculated in the case of the age-class transition matrix (see Chapter 10, p. 82-83). has to In that case, the stable age-class vector be determined as the fixed point (eigenvector to the eigenvalue 1) of the matrix PT, i.e., as the solution to . Because the fixed point is attracting, it can be obtained as the limit of the sequence r starting from an initial vector a0 . 111 ,